More generally, the equivalence \(a^*\)  sends the cone stack associated to a combinatorial cone stack \(\Psi :\mathcal {C}\rightarrow \operatorname{\bf {RPC}}^f\)  (in the sense of [1]}) to the Artin fan which is equal to the following colimit in the category of logarithmic algebraic stacks
\(\varinjlim _{C\in \mathcal {C}} \big [\operatorname{Spec}k[S_{\Psi (C)}]\big /\operatorname{Spec}k[S_{\Psi (C)}^{gp}]\big ],\) 
